Bladder disorders affect millions of Americans, disrupting sleep, work and everyday life and, in some cases, leaving people unable to fully control when they urinate. They also contribute to substantial health care costs.
Researchers still don’t fully understand why the lower urinary tract stops functioning normally or which part of the system should be targeted to restore that function.
University of Maine researchers believe mathematics could help answer those questions. But first, they have to invent some of the math they need.
The bladder fills gradually, then empties abruptly. That sudden switch is central to healthy function, but it is also the kind of behavior many conventional mathematical tools struggle to describe accurately.
Rather than ignore these sudden transitions or replace them with gradual ones to make the problem easier to solve, UMaine researchers are developing new mathematics that stays true to what the body is actually doing.
Peter Stechlinski, associate professor of mathematics, and Giovanna Guidoboni, vice president for research, are leading the U.S. National Science Foundation-supported project. Their work is advancing the theory of nonsmooth dynamical systems while using the lower urinary tract as a real-world test case.
The researchers aim to develop methods that can show how changes in individual parts of a system affect its overall cycle and identify the conditions that allow those cycles to remain stable.
“Because this problem is inherently nonsmooth, we can’t rely on standard mathematical techniques developed for smooth systems,” Stechlinski said. “Those methods may give us inaccurate insights and predictions.”
Stechlinski is principal investigator on the project and Guidoboni is co-principal investigator. Rajat Rai, a UMaine Ph.D. student, also contributed significantly to developing the project and its proposal.
“We have to develop new mathematical tools in order to be able to actually help people,” Guidoboni said.
The lower urinary tract depends on a coordinated system of organs, muscles and neural signals. As the bladder fills, mechanisms controlling the release of urine remain closed. When it is time to urinate, the system rapidly switches states and releases urine.
Those abrupt transitions present a mathematical challenge. Many conventional methods are designed for systems that change smoothly. Researchers can replace a sudden transition with a smoother approximation, but doing so can remove a defining feature of the biological system.
“We don’t want to make it smooth,” Guidoboni said.
For Rai, that decision is central to the project. Instead of changing the representation of the urinary system to fit mathematics that already exists, the researchers are allowing biology to dictate the mathematics they need.
“We’re trying to stay true to the actual system,” Rai said. “The system demands new mathematics, so we choose to develop it rather than simplify the representation of the biology just to make the problem easier to solve.”
The researchers aim to create new ways to analyze sudden transitions, including methods for determining how sensitive a cycle is to changes within the system and whether it remains stable as conditions change.
Bladder function depends on interconnected factors, including neural control, muscle contractions and the mechanics of multiple organs. A disruption in one part can alter the behavior of the entire system, making it difficult to determine from symptoms alone what has gone wrong.
“Our work will study all these connections and provide a quantitative guide as to where to intervene, to regain function,” Guidoboni said.
One focus is voiding efficiency, or how effectively the bladder empties. The researchers want to determine what allows normal cycles of filling and emptying to occur and what causes those cycles to break down.
“We want to determine which factors are driving dysfunction and whether the system’s cycles remain stable over time,” Stechlinski said. “Answering those questions requires us to develop new sensitivity and stability theory for nonsmooth dynamical systems.”
Lower urinary tract research often relies on animal studies to investigate the causes of symptoms and dysfunction. Such studies can be costly, invasive and time-consuming.
The UMaine researchers will test their mathematical predictions against an extensive database of animal-study data to determine whether the models can reliably reproduce bladder behavior already observed experimentally.
If they can, researchers could use the models to investigate new hypotheses computationally and help identify which questions or potential interventions warrant further laboratory study.
The project also challenges the traditional divide between theoretical and applied mathematics. The researchers are not simply applying established mathematics to a medical problem. The medical problem itself is exposing a need for new mathematical tools.
“We don’t value the theory more than the application or vice versa,” Guidoboni said. “We value them individually and in connection.”
The project brings together expertise in mathematics, engineering and lower urinary tract physiology and includes collaboration with associate professor Zachary Danziger from Emory University School of Medicine.
Its mathematical reach could extend beyond the bladder. Rai noted that nonsmooth behavior can arise when parts of a system operate on very different timescales. In the urinary tract, for example, neural signals occur rapidly while the bladder fills over a much longer period.
Guidoboni pointed to the heart as another example. Like the bladder, it repeatedly fills and empties while valves rapidly open and close. New methods for analyzing these systems could have applications across biology, medicine and engineering.
For people living with bladder dysfunction, the goal is more immediate. By developing mathematics capable of showing why the body’s normal cycle of storing and releasing urine breaks down, the researchers hope to give scientists a clearer picture of where to look for ways to restore it.
Contact: David Nordman, david.nordman@maine.edu
